Method of temporal bipartite projection

ABSTRACT

The present disclosure relates, according to some embodiments, to a method of temporal bipartite projection for users and objects and a method of link prediction for an unhappened event. The method of the temporal bipartite projection comprises making a sequence of user-object weighted bipartite networks with the user-object weights, identifying the transitions for each user from a first object at a first time to a second object at a second time, assigning the transition weights corresponding to the transitions according to a predetermined rule, summing the transition weights for all users between two objects to obtain the transition tendencies, and constructing a temporal projection graph with the transition tendencies. The method of link prediction comprises constructing the temporal projection graph, identifying the potential transitions for the unhappened event at a target time, assigning the potential transition weights according to a second predetermined rule, summing the potential transition weights to obtain the scores, and ranking the scores of the link prediction.

FIELD OF THE DISCLOSURE

The present disclosure relates, according to some embodiments, to methods of temporal bipartite projection. For example, some embodiments of the present disclosure provides for methods of temporal bipartite projection to generate temporal projection graphs that characterize the transition tendencies among objects.

BACKGROUND OF THE DISCLOSURE

Online social networks are becoming increasingly popular and may be very dynamic and engaging for users. Many activities may occur in such online social networks: people may interact with one another, express opinions, share files, and view or purchase products or services from all over the world. Online social networks may be modeled by sequences of bipartite networks indexed in time. Bipartite networks may use two types of nodes: users and objects. Object nodes may represent things such as products, files, articles, and author-conference datasets. Such bipartite networks are called user-object networks.

A traditional method for bipartite projection projects a bipartite graph into a unipartite graph of one type of node from the original bipartite graph. For user-object networks, the traditional method projects the user-object graph into either a user graph or an object graph. FIG. 1 depicts a schematic diagram of a traditional bipartite projection method. As seen in FIG. 1, a user-object bipartite network G_(UO) has user nodes U={u₁, u₂, . . . u₇} and object nodes O={o_(1, 2), . . . o₅}. The user-object bipartite network G_(UO) is projected into a user network G_(U) and an object network G_(O). Accordingly, if two nodes in the user-object graph have common neighbors, projected networks G_(U) and G_(O) may be established through a user projection P_(U) and an object projection P_(O), respectively.

The traditional bipartite projection method maps the unweighted user-object bipartite network G_(UO) to either the user network G_(U) or the object network G_(O). Accordingly, one has to collapse all the temporal bipartite networks into a single network in order to make use of such a method. This results in the loss of valuable temporal information and individual user history. Without the temporal information, it may be very difficult to find out whether an object is the substitute of another object.

Therefore, there is a need for a method of temporal bipartite projection that takes into account temporal information.

SUMMARY OF THE DISCLOSURE

The present disclosure describes methods of temporal bipartite projection and application methods thereof. The present disclosure relates, in some embodiments, to methods of temporal bipartite projection that generate temporal projection graphs by mapping sequences of weighted bipartite networks to weighted directed projected object graphs by assigning corresponding transition tendencies as its edge weights.

In some embodiments, a method of temporal bipartite projection for users and objects is provided. The method comprises receiving a user-object dataset, wherein the user-object dataset comprises a set of users, a set of objects, and a set of times for which the events between the users and the objects occur at; making a sequence of user-object weighted bipartite networks {G^(t), t=1, 2, . . . , T} with a set of user-object weights between each pair of an n^(th) user and a m^(th) object at a time t by data processing to the user-object dataset; identifying a set of transitions R(n, i, t₁, j, t₂) for the n^(th) user from a i^(th) object at a first time t₁ to a j^(th) object at a second time t₂ according to the user-object weighted bipartite networks G^(t), wherein t₂>t₁; assigning a set of transition weights w(n, i, t₁, j, t₂) corresponding to the transitions R(n, i, t₁, j, t₂) according to a predetermined rule; summing the transition weights w(n, i, t₁, j, t₂) for all users to obtain a set of transition tendencies ĝ_(i,j) from the i^(th) object to the j^(th) object; and constructing a temporal projection graph Ĝ for all objects with the transition tendencies ĝ_(i,j).

In some embodiments, a method of generating a popularity index for objects is provided. The method comprises constructing the temporal projection graph Ĝ in an adjacency matrix form; estimating a first accumulated number of non-zero entries in the i^(th) column of the temporal projection graph Ĝ as an indegree d_(in)(i) of the i^(th) object O_(i) and a second accumulated number of non-zero entries in the i^(th) row of the temporal projection graph Ĝ as an outdegree d_(out)(i) of the i^(th) object O_(i); and calculating a ratio of the indegree d_(in)(i) to the sum the indegree d_(in)(i) and the outdegree d_(out)(i) as the popularity index Popul(i) of the i^(th) object O_(i).

In some embodiments, a method of estimating a transition probability for objects is provided. The method comprises constructing a temporal projection graph Ĝ in an adjacency matrix form; summing all entries in the i^(th) column of the temporal projection graph Ĝ; and calculating a ratio of the transition tendency ĝ_(i,j) to the sum of all entries in the i^(th) column of the temporal projection graph Ĝ as the transition probability p_(i,j) for all users from the i^(th) object O_(i) to the j^(th) object O_(j).

In some embodiments, a method of link prediction for an unhappened event between the users and objects. The method comprises constructing the temporal projection graph Ĝ in an adjacency matrix form; identifying a set of potential transitions R_(p)(n, i, t₃, j, T+1) for the unhappened events of the n^(th) user from the i^(th) object O_(i) at a third time t₃ to the j^(th) object O_(j) at a target time T+1, wherein 1≦t₃≦T, and 1≦i, j≦M assigning a set of potential transition weights w_(p)(n, i, t₃, j, T+1) corresponding to the potential transitions R_(p)(n, i, t₃, j, T+1) according to a second predetermined rule; summing the potential transition weights w_(p)(n, i, t₃, j, T+1) for the n^(th) user and the j^(th) object O_(j) to obtain a set of scores

${{{Score}\left( {n,j} \right)} = {\sum\limits_{t_{3} = 1}^{T}{\sum\limits_{i = 1}^{M}{w_{p}\left( {n,i,t_{3},j,{T + 1}} \right)}}}};$

and ranking the scores Score(n,j) for the link prediction of the unhappened events of the n^(th) user and the j^(th) object O_(j).

Therefore, the present disclosure provides for methods of temporal bipartite projection and application methods thereof that take into account the temporal information for identifying substitute relationships between two objects by assigning transition weights for the transitions. Accordingly, the transition tendencies obtained this way can be viewed as a social aggregated change of the preferences among all objects.

The foregoing is a summary and shall not be construed to limit the scope of the claims. The operations and devices disclosed herein may be implemented in a number of ways, and such changes and modifications may be made without departing from this disclosure and its broader aspects. Other aspects, inventive features, and advantages of the disclosure, as defined solely by the claims, are described in the non-limiting detailed description set forth below.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a schematic diagram of a traditional bipartite projection;

FIG. 2A illustrates an example of a user-object dataset for a method of temporal bipartite projection in accordance with one embodiment of the present disclosure;

FIG. 2B illustrates an example of a sequence of user-object weighted bipartite networks for a method of temporal bipartite projection in accordance with one embodiment of the present disclosure;

FIG. 2C illustrates an example of a set of transition weights and a temporal projection graph for a method of temporal bipartite projection in accordance with one embodiment of the present disclosure;

FIG. 2D illustrates a flow chart of a method of temporal bipartite projection in accordance with one embodiment of the present disclosure;

FIG. 3 illustrates a flow chart of a method of generating a popularity index for objects in accordance with one embodiment of the present disclosure;

FIG. 4 illustrates a flow chart of a method of estimating a transition probability for objects in accordance with one embodiment of the present disclosure; and

FIG. 5 illustrates a flow chart of a method of link prediction for an unhappened event in accordance with one embodiment of the present disclosure.

DETAILED DESCRIPTION OF EMBODIMENTS

Reference will be made to example embodiments illustrated in the accompanying drawings. Wherever possible, the same reference numbers are used in the drawings and the description to refer to the same or like parts.

FIG. 2A illustrates an example of a user-object dataset for a method of temporal bipartite projection. FIG. 2B illustrates an example of a sequence of user-object weighted bipartite networks for a method of temporal bipartite projection. FIG. 2C illustrates an example of a set of transition weights and a temporal projection graph for a method of temporal bipartite projection. FIG. 2D illustrates a flow chart of a method of temporal bipartite projection in accordance with one embodiment of the present disclosure.

The present disclosure relates, in some embodiments, to a method of temporal bipartite projection. According to the method, first, a computer may receive a user-object dataset D comprising a plurality of data d₁-d₄, in which each data (d₁, d₂, d₃, or d₄) includes users U_(k), objects O_(p), and occurred times a_(t) related between the users and the objects (S201). As shown in FIG. 2A, the method may take a conference dataset as an example for the user-object dataset D, takes the authors as the users U_(k), take the conferences as the objects O_(p), and takes the publishing years as the occurred times a_(t), where k=1, N; p=1, 2, . . . , M; and t=1, 2, . . . , T, where a_(T) indicates a training period. One of ordinary skill in the art, having the benefit of the present disclosure, would appreciate that the users and objects are not limited to the authors and the conferences, other pairs of users and objects might be also applied.

Next, according to the method, the computer may operate a data processing to the user-object dataset D, such as classification by time, user, and object, to making a sequence of user-object weighted bipartite networks {G^(t), t=1, 2, . . . , T} in an adjacency matrix form with a set of user-object weights g_(n,m) ^(t) as shown in FIG. 2B (S203). Each of the user-object weights g_(n,m) ^(t) in n^(th) column and m^(th) row of the user-object weighted bipartite networks G^(t) indicates a weight (link times) between each pair of a n^(th) user U_(n) and a m^(th) object O_(m) at a time a_(t). For example, the value of g_(1,1) ¹ is 1 indicates that the first user U₁ had been submitted one journal to the first conference O₁ at time a₁, and the value of g_(2,1) ¹ is 2 indicates the second author U₂ had been submitted two journals to the first conference O₁ at time a₁.

The computer may identify a set of transitions R(n, i, t₁, j, t₂) for each user U_(n) from a i^(th) object O_(i) at a first time t₁ to a j^(th) object O_(j) at a second time t₂ according to the user-object weighted bipartite networks G^(t), wherein 1≦n≦N, 1≦t₁≦t₂≦T, and 1≦i, j≦M (S205). As shown in FIG. 2C, the transition R_(a)(1, 1, 1, 2, 2) indicates that the first user U₁ contributed to the first object O₁ at time a₁, and then contributed to the second object O₂ at time a₂.

Then, according to the method, the computer may assign a set of transition weights w(n, i, t₁, j, t₂) for the transitions R(n, i, t₁, j, t₂) respectively according to a predetermined rule for obtaining transition tendencies between two objects (S207).

Then, according to the method, the computer may sum the transition weights w(n, i, t₁, j, t₂) for all users U_(k) between two objects O_(i), O_(j), to obtain a set of transition tendencies

$\begin{matrix} {{\hat{g}}_{i,j} = {\sum\limits_{n = 1}^{N}{\sum\limits_{t_{1} = 1}^{T - 1}{\sum\limits_{t_{2} = {t_{1} + 1}}^{T}{{w\left( {n,i,t_{1},j,t_{2}} \right)}.}}}}} & ({S209}) \end{matrix}$

Lastly, according to the method, the computer may construct a temporal projection graph Ĝ for all objects with the transition tendencies ĝ_(i,j) (S211). More specifically, each of the transition tendencies ĝ_(i,j) denotes the entry of the i^(th) column and the j^(th) row of the temporal projection graph Ĝ in an adjacency form.

In some embodiments, the predetermined rule comprises that each of the transition weights w(n, i, t₁, j, t₂) is increasing in the first time t₁ at which the transition R(n, i, t₁, j, t₂) occurs. For example, a recent transition should carry a larger transition weight than an obsolete transition. Therefore, the computer may identify a first weighting factor α₁ (t₁) for each of the transition weights w(n, i, t₁, j, t₂) that is increasing in t₁.

In some embodiments, the predetermined rule comprises that each of the transition weights w(n, i, t₁, j, t₂) is increasing in the user-object weight g_(n,i) ^(t) ¹ between the n^(th) user U_(n) and the i^(th) object O_(i) at the first time t₁. For example, a transition with a larger user-object weight g_(n,i1) ^(t) ¹ between a user U_(n) and an object O₁₁ at a time t₁ should carry a larger transition weight than a transition with a smaller weight g_(n,i2) ^(t) ¹ between the same user and another object O_(i2) at the same time t₁. Therefore, the computer may define a second weighting factor α₂ (g_(n,i) ^(t) ¹ ) for each of the transition weights w(n, i, t₁, j, t₂) that is increasing in the user-object weight g_(n,i) ^(t) ¹ .

In some embodiments, the predetermined rule comprises that each of the transition weights w(n, i, t₁, j, t₂) is decreasing in the duration (t₂−t₁) of the transition R(n, i, t₁, j, t₂). For example, a transition with a shorter time difference should carry a larger transition weight than a transition with a larger time difference. Therefore, the computer may define a third weighting factor α₃ (t₂−t₁) for each of the transition weights w(n, i, t₁, j, t₂) that is decreasing in the duration (t₂−t₁) of the transition R(n, i, t₁, j, t₂).

In some embodiments, the predetermined rule comprises that the transition R(n, i, t₁, j, t₂) for each of the users U_(k) should be normalized. Since each of the transition tendencies may be viewed as a social aggregated preference of the objects O_(p), the assignment of the transition weights w(n, i, t₁, j, t₂) should avoid one user U_(n) dominating the social aggregated preference. For example, if a user U_(n) has a transition R(n, i, t₁, j, t₂) for a very small t₁, there may be many transitions for the user U_(n) in a sequence of user-object weighted bipartite networks G^(t). Therefore, the computer may define a fourth weighting factor α₄ (n, t₁) for normalizing each of the transition weights w(n, i, t₁, j, t₂) of each user U_(n).

In some embodiments, the computer may use the simple product of these four weighting factors α₁, α₂, α₃, α₄ for the transition weights w(n, i, t₁, j, t₂) to satisfy the following equation:

w(n,i,t ₁ ,j,t ₂)=α₁(t ₁)×α₂(g _(n,i) ^(t) ¹ )×α₃(t ₂ −t ₁)×α₄(n,t ₁)  (1)

To simplify the calculation, the computer may assume that the fourth weighting factor α₄(n, t₁) is determined by the second weighting factor α₂ and the third weighting factor α₃. Accordingly, the weighting factors satisfy the following equation:

$\begin{matrix} {{\alpha_{4}\left( {n,t_{1}} \right)} = \frac{1}{\left( {\sum\limits_{i = 1}^{M}{{\alpha_{2}\left( g_{n,i}^{t_{1}} \right)}\left( {\sum\limits_{s = t_{1}}^{T}{I_{n}^{s} \times {\alpha_{3}\left( {t_{2} - t_{1}} \right)}}} \right)}} \right.}} & (2) \end{matrix}$

In equation (2), the computer may assume that I_(n) ^(t) denotes an indicator variable for the event with positive user-object weight g_(n,i) ^(t). Accordingly, I_(n) ^(t) satisfies the following equation:

$\begin{matrix} {I_{n}^{t} = \left\{ \begin{matrix} {1,{{{if}\mspace{14mu} {\sum\limits_{j = 1}^{M}g_{n,j}^{t}}} > 0}} \\ {0,{otherwise}} \end{matrix} \right.} & (3) \end{matrix}$

The transition weights w(n, i, t₁, j, t₂) may further satisfy the following equation:

$\begin{matrix} {{w\left( {n,i,t_{1},j,t_{2}} \right)} = {{\alpha_{1}\left( t_{1} \right)} \times \frac{\alpha_{2}\left( g_{n,i}^{t_{1}} \right)}{\sum\limits_{l = 1}^{m}{\alpha_{2}\left( g_{n,l}^{t_{1}} \right)}} \times \frac{\alpha_{3}\left( {t_{2} - t_{1}} \right)}{\sum\limits_{s = {t_{1} + 1}}^{T}{I_{n}^{s} \times {\alpha_{3}\left( {s - t_{1}} \right)}}}}} & (4) \end{matrix}$

Further, the computer may simplify the transition weights w(n, i, t₁, j, t₂) by assuming that α₁(t)=α₃(T−t)=α(t) and α₂(g_(n,i) ^(t) ¹ )=g_(n,i) ^(t) ¹ . Accordingly, the transition weights w(n, i, t₁, j, t₂) satisfy the following equation:

$\begin{matrix} {{w\left( {n,i,t_{1},j,t_{2}} \right)} = {{\alpha \left( t_{1} \right)} \times \frac{g_{n,i}^{t_{1}}}{\sum\limits_{l = 1}^{M}g_{n,l}^{t_{1}}} \times \frac{\alpha \left( {T - \left( {t_{2} - t_{1}} \right)} \right)}{\sum\limits_{s = {t_{1} + 1}}^{T}{I_{n}^{s} \times {\alpha \left( {T - \left( {s - t_{1}} \right)} \right)}}}}} & (5) \end{matrix}$

Therefore, the transition tendencies ĝ_(i,j) may satisfy the following equation:

$\begin{matrix} {{\hat{g}}_{i,j} = {\sum\limits_{n = 1}^{N}{\sum\limits_{t_{1} = 1}^{T - 1}{\sum\limits_{t_{2} = {t_{1} + 1}}^{T}\left( {{\alpha \left( t_{1} \right)} \times \frac{g_{n,i}^{t_{1}}}{\sum\limits_{l = 1}^{M}g_{n,l}^{t_{1}}} \times \frac{\alpha \left( {T - \left( {t_{2} - t_{1}} \right)} \right)}{I_{n}^{s} \times {\alpha \left( {T - \left( {s - t_{1}} \right)} \right)}}} \right)}}}} & (6) \end{matrix}$

In equation (6), the computer may assume that I_(n,j) ^(t) ² =1 if g_(n,j) ^(t) ² >0, and I_(n,j) ^(t) ² =0 otherwise. Note that only if there is a transition R(n, i, t₁, j, t₂), g_(n,i) ^(t) ¹ >0, and I_(n,j) ^(t) ² =1.

In some embodiments, the computer may assume that α(t)=(0.8)^(T-t). To take the transition tendency ĝ_(1,2) of the first user U₁ from the first object O₁ to the second object O₂ for example, ĝ_(1,2) may be calculated through equation (6) according to the transition R_(a)(1,1, 1, 2, 2), transition R_(b)(1, 1, 1, 2, 3) and the user-object weight g_(1,1) ¹ thereof with the following equation:

$\begin{matrix} {{\hat{g}}_{1,2} = {{0.8^{3 - 1}\frac{1}{1 + 1}\frac{0.8^{3 - {({3 - 1})}}}{0.8 + 0.8^{2}}} + {0.8^{3 - 1}\frac{1}{1 + 1}\frac{0.8^{3 - {({3 - 2})}}}{0.8 + 0.8^{2}}}}} \\ {= {0.18 + 0.14}} \\ {= 0.32} \end{matrix}$

Accordingly, the temporal projection graph Ĝ in an adjacency form may be constructed as shown in FIG. 2C and stored in the memory of the computer.

FIG. 3 illustrates a flow chart of a method of generating a popularity index for objects in accordance with some embodiments of the present disclosure. According to the method of generating a popularity index for objects, first, the computer may construct the temporal projection graph Ĝ in an adjacency matrix form according to the above-mentioned embodiments (S301). Then, the computer may estimate a first accumulated number of non-zero entries in the i^(th) column of the temporal projection graph Ĝ as an indegree d_(in)(i) of the i^(th) object O_(i) and a second accumulated number of non-zero entries in the i^(th) row of the temporal projection graph Ĝ as an outdegree d_(out)(i) of the i^(th) object O_(i) (S303). Lastly, the computer may calculate a ratio of the indegree d_(in)(i) to the sum of the indegree d_(in)(i) and the outdegree d_(out)(i) as the popularity index Popul(i) for the i^(th) object O_(i) (S305).

The popularity index Popul(i) for each of the objects o_(p) may fall between 0 and 1. Moreover, if d_(in)(i)<<d_(out)(i), then Popul(i)→1. For the i^(th) object O_(i) with a high popularity index Popul(i), one may infer that this object O_(i) is popular recently because there are many users U_(n) with links to other objects that have made transition to this object O_(i). On the other hand, if d_(in)(i)>>d_(out)(i), then Popul(i)→0. For the i^(th) object O_(i) with a low popularity index Popul(i), one may infer that this object O_(i) was popular in the past and unpopular now because there are many users U_(n) with links to this object O_(i) that have made transition to other objects.

FIG. 4 illustrates a flow chart of a method of estimating a transition probability for objects in accordance with some embodiments of the present disclosure. The method of estimating a transition probability for objects comprises the steps as follows: First, the computer may construct the temporal projection graph Ĝ in an adjacency matrix form according to the above-mentioned embodiments (S401). Then, the computer may sum all entries in the i^(th) column of the temporal projection graph Ĝ (S403). Lastly, the computer may calculate a ratio of the transition tendency ĝ_(i,j) to the sum of all entries in the i^(th) column of the temporal projection graph Ĝ as the transition probability p_(i,j) for all users U_(k) from the i^(th) object O_(i) to the j^(th) object O_(j) (S409).

FIG. 5 illustrates a flow chart of a method of link prediction for an unhappened event in accordance with some embodiments of the present disclosure. First the computer may construct the temporal projection graph Ĝ in an adjacency matrix form according to the above-mentioned embodiments (S501). Then, the computer may identify a set of potential transitions R_(p)(n, i, t₃, j, T+1) for the unhappened events of the users U_(k) from the i^(th) object O_(i) at a third time t₃ to the j^(th) object O_(j) at a target (or future) time T+1, where 1≦n≦N, 1≦t₃≦T, and 1≦i, j≦M.

Next, the computer may identify a set of potential transitions R_(p)(n, i, t₃, j, T+1) for the unhappened events of the users U_(k) from the i^(th) object O_(i) at a third time t₃ to the j^(th) object O_(j) at a target time T+1 (S503).

Next, the computer may assign a set of potential transition weights w_(p)(n, i, t₃, j, T+1) corresponding to the potential transitions R_(p)(n, i, t₃, j, T+1) according to a second predetermined rule (S505).

Next, the computer may sum the potential transition weights w_(p)(n, i, t₃, j, T+1) for each of the users U_(k) and the j^(th) object O_(j) to obtain a set of scores

$\begin{matrix} {{{Score}\left( {n,j} \right)} = {\sum\limits_{t_{3} = 1}^{T}{\sum\limits_{i = 1}^{M}{{w_{p}\left( {n,i,t_{3},j,{T + 1}} \right)}.}}}} & ({S507}) \end{matrix}$

Lastly, the computer may rank the scores Score(n, j) of the link prediction for the unhappened events of the users U_(k) and the j^(th) object O_(j) (S509).

In some embodiments, the second predetermined rule comprises that each of the potential transition weights w_(p)(n, i, t₃, j, T+1) is increasing in the third time t₃ at which the potential transition R_(p)(n, i, t₃, j, T+1) occurs. Similar to the first weighting factor α₁, the computer may identify a fifth weighting factor α₅(t₃) for each of the potential transition weights w_(p)(n, i, t₃, j, T+1) that is increasing in t₃.

In some embodiments, the second predetermined rule comprises that each of the potential transition weights w_(p)(n, i, t₃, j, T+1) is increasing in the user-object weight g_(n,i) ^(t) ⁵ between the n^(th) user U_(n) and the i^(th) object O_(i) at the third time t₃. Similar to the second weighting factor α₂, the computer may identify a sixth weighting factor α₆(g_(n,i) ^(t) ⁵ ) for the potential transition weights w_(p)(n, i, t₃, j, T+1) that is increasing in the user-object weight g_(n,i) ^(t) ³ .

In some embodiments, the second predetermined rule comprises that each of the potential transition weights w_(p)(n, i, t₃, j, T+1) is decreasing in the duration (T+1−t₃) of the potential transition R_(p)(n, i, t₃, j, T+1). Similar to the third weighting factor α₃, the computer may identify a seventh factor α₇ (T+1−t₃) for the potential transition weights w_(p)(n, i, t₃, j, T+1) that is decreasing in the duration (T+1−t₃) of the potential transition R_(p)(n, i, t₃, j, T+1).

In some embodiments, the second predetermined rule comprises each potential transition weight w_(p)(n, i, t₃, j, T+1) is increasing in the transition probability p_(i,j). For example, one potential transition R_(p) with a larger transition probability from one object to the target object should carry a larger transition weight than that with a smaller transition probability from another object to the target object. Therefore, the computer may identify a eighth factor α₈ (p_(i,j)) for each of the potential transition weights w_(p)(n, i, t₃, j, T+1) that is increasing in the transition probability p_(i,j).

According to some embodiments of the present disclosure, the computer may use the simple product of these four weighting factors α₅, α₆, α₇, α₈ for each of the potential transition weights w_(p)(n, i, t₃, j, T+1) to satisfy the following equation:

w _(p)(n,i,t ₃ ,j,T+1)=α₅(t ₃)×α₆(g _(n,i) ^(t) ³ )×α₇(T+1−t ₃)×a ₈(p _(i,j))  (7)

To simply calculations, the computer may assume that these four weighting factors α₅, α₆, α₇, α₈ satisfy the following equations:

α₅(t ₃)×α₇(T+1−t ₃)=α(t)  (8)

α₆(g _(n,i) ^(t) ³ )=g _(n,i) ^(t) ³   (9)

α₈(p _(i,j))=p _(i,j)  (10)

The potential transition weights w_(p)(n, i, t₃, j, T+1) may further satisfy the following equation:

w _(p)(n,i,t ₃ ,j,T+1)=α(t)×g _(n,i) ^(t) ³ ×p _(i,j)  (11)

The scores

${{Score}\left( {n,j} \right)} = {\sum\limits_{t_{3} = 1}^{T}{\sum\limits_{i = 1}^{M}{w_{p}\left( {n,i,t_{3},j,{T + 1}} \right)}}}$

may satisfy the following equation:

$\begin{matrix} {{{Score}\left( {n,j} \right)} = {\sum\limits_{t_{3} = 1}^{T}{\sum\limits_{i = 1}^{M}{{\alpha (t)} \times \delta_{n,i}^{t_{3}} \times p_{i,j}}}}} & (12) \end{matrix}$

One of ordinary skill in the art having the benefit of the present disclosure would appreciate that the transition probability p_(i,j) may be stored in the memory of the computer in advance. In some embodiments, the computer may assume that α(t)=(0.8)^(T-t). Thus the scores Score(n, j) for all users U_(k) may be calculated as following equations:

Score(1,1)=α(1)×p _(1,1)≅0.24  (13)

Score(1,2)=α(1)×g _(1,1) ¹ ×p _(1,2)+α(1)×g _(1,2) ¹ ×p _(2,2)+α(2)×g _(1,2) ² ×p _(2,2)+α(3)×g _(1,2) ³ ×p _(2,2)≅1.87  (14)

Score(1,3)=α(1)×g _(1,1) ¹ ×p _(1,3)+α(1)×g _(1,2) ¹ ×p _(2,3)+α(2)×g _(1,2) ² ×p _(2,3)+α(3)×g _(1,2) ³ ×p _(2,3)≅1.78  (15)

Score(2,1)=α(1)×g _(2,1) ¹ ×p _(1,1)+α(3)×g _(2,1) ³ ×p _(1,1)≅0.84  (16)

Score(2,2)=α(1)×g _(2,1) ¹ ×p _(1,2)+α(3)×g _(2,1) ³ ×p _(1,2)≅0.41  (17)

Score(2,3)=α(1)×g _(2,1) ¹ ×p _(1,3)+α(3)×g _(2,1) ³ ×p _(1,3)≅1.03  (18)

Accordingly, the computer may sort the numerical results to rank these user-object pairs for link prediction. The results indicate that the pair of the first user U₁ and the second object O₂ has the highest score.

The present disclosure may advantageously improve the ability to provide a method of temporal bipartite projection for users and objects to obtain the temporal projection graph that represents a set of transition tendencies among objects over time. Based on the temporal projection graph, the popularity index of objects for ranking objects and the transition probability between two objects are obtained. Additionally, the score for temporal prediction may rank the likelihood of the occurrence of a future link between a user and an object.

Realizations in accordance with the present disclosure have been described in the context of particular embodiments. These embodiments are meant to be illustrative and not limiting. Many variations, modifications, additions, and improvements are possible. Accordingly, plural instances may be provided for components described herein as a single instance. Structures and functionality presented as discrete components in the exemplary configurations may be implemented as a combined structure or component. These and other variations, modifications, additions, and improvements may fall within the scope of the disclosure as defined in the claims that follow.

While various embodiments in accordance with the principles disclosed herein have been described above, it should be understood that they have been presented by way of example only, and not limitation. Thus, the breadth and scope of this disclosure should not be limited by any of the above-described exemplary embodiments, but should be defined only in accordance with any claims and their equivalents issuing from this disclosure. Furthermore, the above advantages and features are provided in described embodiments, but shall not limit the application of such issued claims to processes and structures accomplishing any or all of the above advantages.

Additionally, the section headings herein are provided for consistency with the suggestions under 37 CFR 1.77 or otherwise to provide organizational cues. These headings shall not limit or characterize the embodiment(s) set out in any claims that may issue from this disclosure. Specifically and by way of example, although the headings refer to a “Field of Disclosure,” the claims should not be limited by the language chosen under this heading to describe the so-called field. Further, a description of a technology in the “Background of the Disclosure” is not to be construed as an admission that certain technology is prior art to any embodiment(s) in this disclosure. Neither is the “Summary of the Disclosure” to be considered as a characterization of the embodiment(s) set forth in issued claims. Furthermore, any reference in this disclosure to “invention” in the singular should not be used to argue that there is only a single point of novelty in this disclosure. Multiple embodiments may be set forth according to the limitations of the multiple claims issuing from this disclosure, and such claims accordingly define the embodiment(s), and their equivalents, that are protected thereby. In all instances, the scope of such claims shall be considered on their own merits in light of this disclosure, but should not be constrained by the headings set forth herein. 

What is claimed is:
 1. A method of temporal bipartite projection for users and objects applied, the method comprising: receiving, at a computer, a user-object dataset, wherein the user-object dataset comprises a set of users, a set of objects, a set of times for which the events between the users and the objects occur; generating, at the computer, a sequence of user-object weighted bipartite networks {G^(t), t=1, 2, . . . , T} with a set of user-object weights g_(n,m) ^(t) between each pair of an n^(th) user and a m^(th) object at a time t by data processing the user-object dataset; identifying, at the computer, a set of transitions R(n, i, t₁, j, t₂) for the n^(th) user from i^(th) an object at a first time t₁ to a j^(th) object at a second time t₂ according to the user-object weighted bipartite networks G^(t), wherein t₂>t₁; assigning, at the computer, a set of transition weights w(n, i, t₁, j, t₂) corresponding to the transitions R(n, i, t₁, j, t₂) according to a predetermined rule; summing, at the computer, the transition weights w(n, i, t₁, j, t₂) for all users to obtain a set of transition tendencies ĝ_(i,j) from the i^(th) object and the j^(th) object; and constructing, at the computer, a temporal projection graph Ĝ for all objects with the transition tendencies ĝ_(i,j).
 2. The method of temporal bipartite projection according to claim 1, wherein the predetermined rule comprises each of the transition weights w(n, i, t₁, j, t₂) is increasing in the first time t₁ at which the transition R(n, i, t₁, j, t₂) occurs.
 3. The method of temporal bipartite projection according to claim 1, wherein the predetermined rule comprises each of the transition weights w(n, i, t₁, j, t₂) is increasing in the user-object weight g_(n,i) ^(t) ¹ between the n^(th) user and the i^(th) object at the first time t₁.
 4. The method of temporal bipartite projection according to claim 1, wherein the predetermined rule comprises each of the transition weights w(n, i, t₁, j, t₂) is decreasing in the duration (t₂−t₁) of the transition R(n, i, t₁, j, t₂).
 5. The method of temporal bipartite projection according to claim 1, wherein the predetermined rule comprises the transition R(n, i, t₁, j, t₂) for the n^(th) user should be normalized.
 6. The method of temporal bipartite projection according to claim 1, wherein each of the transition weights w(n, i, t₁, j, t₂) is determined by the equation: w(n,i,t ₁ ,j,t ₂)=α₁(t ₁)×α₂(g _(n,i) ^(t) ¹ )×α₃(t ₂ −t ₁)×α₄(n,t ₁), wherein α₁, α₂, α₃, α₄ are weighting factors, wherein α₁ is increasing in the first time t₁ at which the transition R(n, i, t₁, j, t₂) occurs; α₂ is increasing in the user-object weight g_(n,i) ^(t) ¹ between the n^(th) user and the i^(th) object at the first time t₁; α₃ is decreasing in the duration (t₂−t₁) of the transition R(n, i, t₁, j, t₂); and α₄ is related to normalize each of the transitions R(n, i, t₁, j, t₂).
 7. A method of generating a popularity index for objects, comprising: constructing, at the computer, the temporal projection graph Ĝ according to claim 1 in an adjacency matrix form; estimating, at the computer, a first accumulated number of non-zero entries in the i^(th) column of the temporal projection graph Ĝ as an indegree d_(in)(i) of the i^(th) object and a second accumulated number of non-zero entries in the i^(th) row of the temporal projection graph Ĝ as an outdegree d_(out)(i) of the i^(th) object; and calculating, at the computer, a ratio of the indegree d_(in)(i) to the sum of the indegree d_(in)(i) and the outdegree d_(out)(i) as the popularity index Popul(i) of the i^(th) object.
 8. The method of generating a popularity index for objects according to claim 7, wherein the predetermined rule comprises each of the transition weights w(n, i, t₁, j, t₂) is increasing in the first time t₁ at which the transition R(n, i, t₁, j, t₂) occurs.
 9. The method of generating a popularity index for objects according to claim 7, wherein the predetermined rule comprises each of the transition weights w(n, i, t₁, j, t₂) is increasing in the user-object weight between the n^(th) user and the i^(th) object at the first time t₁.
 10. The method of generating a popularity index for objects according to claim 7, wherein the predetermined rule comprises that each of the transition weights w(n, i, t₁, j, t₂) is decreasing in the duration (t₂−t₁) of the transition R(n, i, t₁, j, t₂).
 11. The method of generating a popularity index for objects according to claim 7, wherein the predetermined rule comprises that the transition R(n, i, t₁, j, t₂) for the n^(th) user should be normalized.
 12. The method of generating a popularity index for objects according to claim 7, wherein each of the transition weights w(n, i, t₁, j, t₂) is determined by the equation: w(n,i,t ₁ ,j,t ₂)=α₁(t ₁)×α₂(g _(n,i) ^(t) ¹ )×α₃(t ₂ −t ₁)×α₄(n,t ₁), wherein α₁, α₂, α₃, α₄ are weighting factors, wherein α₁ is increasing in the first time t₁ at which the transition R(n, i, t₁, j, t₂) occurs; α₂ is increasing in the user-object weight between the n^(th) user and the i^(th) object at the first time t₁; α₃ is decreasing in the duration (t₂−t₁) of the transition R(n, i, t₁, j, t₂); and α₄ is related to normalize each of the transitions R(n, i, t₁, j, t₂).
 13. A method of estimating a transition probability for objects comprising: constructing, at the computer, the temporal projection graph Ĝ according to claim 1 in an adjacency matrix form; summing, at the computer, all entries in the i^(th) column of the temporal projection graph Ĝ; and calculating, at the computer, a ratio of the transition tendency ĝ_(i,j) to the sum of all entries in the i^(th) column of the temporal projection graph Ĝ as the transition probability p_(i,j) for all users from the i^(th) object to the j^(th) object.
 14. The method of estimating a transition probability for objects according to claim 13, wherein the predetermined rule comprises that each of the transition weights w(n, i t₁, j, t₂) is increasing in the first time t₁ at which the transition R(n, i, t₁, j, t₂) occurs.
 15. The method of estimating a transition probability for objects according to claim 13, wherein the predetermined rule comprises that each of the transition weights w(n, i t₁, j, t₂) is increasing in the user-object weight g_(n,i) ^(t) ¹ between the n^(th) user and the i^(th) object at the first time t₁.
 16. The method of estimating a transition probability for objects according to claim 13, wherein the predetermined rule comprises that each of the transition weights w(n, i t₁, j, t₂) is decreasing in the duration (t₂−t₁) of the transition R(n, i, t₁, j, t₂).
 17. The method of estimating a transition probability for objects according to claim 13, wherein the predetermined rule comprises that the transition R(n, i, t₁, j, t₂) for the n^(th) user should be normalized.
 18. The method of estimating a transition probability for objects according to claim 13, wherein each of the transition weights w(n, i, t₁, j, t₂) is determined by the equation: w(n,i,t ₁ ,j,t ₂)=α₁(t ₁)×α₂(g _(n,i) ^(t) ¹ )×α₃(t ₂ −t ₁)×α₄(n,t ₁), wherein α₁, α₂, α₃, α₄ are weighting factors, wherein α₁ is increasing in the first time t₁ at which the transition R(n, i, t₁, j, t₂) occurs; α₂ is increasing in the user-object weight between the n^(th) user and the i^(th) object at the first time t₁; α₃ is decreasing in the duration (t₂−t₁) of the transition R(n, i, t₁, j, t₂); and α₄ is related to normalize each of the transitions R(n, i, t₁, j, t₂).
 19. A method of link prediction for an unhappened event, comprising: constructing, at the computer, the temporal projection graph Ĝ according to claim 1 in an adjacency matrix form; identifying, at the computer, a set of potential transitions R_(p)(n, i, t₃, j, T+1) for the unhappened events of the n^(th) user from the i^(th) object at a third time t₃ to the j^(th) object at a target time T+1, wherein 1≦t₃≦T; assigning, at the computer, a set of potential transition weights w_(p)(n, i, t₃, j, T+1) corresponding to the potential transitions R_(p)(n, i, t₃, j, T+1) according to a second predetermined rule; summing, at the computer, the potential transition weights w_(p) (n, i, t₃, j, T+1) for the n^(th) user and the j^(th) object to obtain a set of scores Score(n,j); and ranking, at the computer, the scores Score(n, j) for the link prediction of the unhappened events of the n^(th) user and the j^(th) object.
 20. The method of link prediction for an unhappened event according to claim 19, wherein the second predetermined rule comprises each of the potential transition weights w_(p)(n, i, t₃, j, T+1) is increasing in the third time t₃ at which the potential transition R_(p)(n, i, t₃, j, T+1) occurs.
 21. The method of link prediction for an unhappened event according to claim 19, wherein the second predetermined rule comprises that each of the potential transition weights w_(p)(n, i, t₃, j, T+1) is increasing in the user-object weight g_(n,i) ^(t) ³ between the n^(th) user and the i^(th) object at the third time t₃.
 22. The method of link prediction for an unhappened event according to claim 19, wherein the second predetermined rule comprises that each of the potential transition weights w_(p)(n, i, t₃, j, T+1) is decreasing in the duration (T+1−t₃) of the potential transition R_(p)(n, i, t₃, j, T+1).
 23. The method of link prediction for an unhappened event according to claim 19, further comprising summing, at the computer, all entries in the i^(th) column of the temporal projection graph Ĝ, and calculating, at the computer, a ratio of the transition tendency ĝ_(i,j) to the sum of all entries in the i^(th) column of the temporal projection graph Ĝ as a transition probability p_(i,j) from the i^(th) object to the j^(th) object.
 24. The method of link prediction for an unhappened event according to claim 23, wherein the second predetermined rule comprises that each of the transition weights w_(p)(n, i, t₃, j, T+1) is increasing in the transition probability p_(i,j).
 25. The method of link prediction for an unhappened event according to claim 19, wherein the predetermined rule comprises that each of the transition weights w(n, i t₁, j, t₂) is increasing in the first time t₁ at which the transition R(n, i, t₁, j, t₂) occurs.
 26. The method of link prediction for an unhappened event according to claim 19, wherein the predetermined rule comprises that each of the transition weights w(n, i t₁, j, t₂) is increasing in the user-object weight g_(n,i) ^(t) ¹ between the n^(th) user and the i^(th) object at the first time t₁.
 27. The method of link prediction for an unhappened event according to claim 19, wherein the predetermined rule comprises that each of the transition weights w(n, i t₁, j, t₂) is decreasing in the duration (t₂−t₁) of the transition R(n, i, t₁, j, t₂).
 28. The method of link prediction for an unhappened event according to claim 19, wherein the predetermined rule comprises that the transition R(n, i, t₁, j, t₂) for the n^(th) user should be normalized.
 29. The method of link prediction for an unhappened event according to claim 19, wherein each of the transition weights w(n, i, t₁, j, t₂) is determined by the equation: w(n,i,t ₁ ,j,t ₂)=α₁(t ₁)×α₂(g _(n,i) ^(t) ¹ )×α₃(t ₂ −t ₁)×α₄(n,t ₁), wherein α₁, α₂, α₃, α₄ are weighting factors, wherein α₁ is increasing in the first time t₁ at which the transition R(n, i, t₁, j, t₂) occurs; α₂ is increasing in the user-object weight g_(n,i) ^(t) ¹ between the n^(th) user and the i^(th) object at the first time t₁; α₃ is decreasing in the duration (t₂−t₁) of the transition R(n, i, t₁, j, t₂); and α₄ is related to normalize each of the transitions R(n, i, t₁, j, t₂). 